On semiperfect $a$-rings - Van, Truong Thi Thuy; Alghamdi, Ahmad M.; Alkinani, Amnah Abdu; Van, Truong Thi Thuy; Alghamdi, Ahmad M.; Alkinani, Amnah Abdu (Institute of Mathematics, NAS of Ukraine, 2024)
UDC 512.5 A ring is called a right $a$-ring if every right ideal is automorphism invariant. We describe some properties of $a$-rings over semiperfect rings. It is shown that an I-finite right $a$-ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if $R$ is an indecomposable (as a ring) I-finite right $a$-ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and ${\rm Soc}(R_R)={\rm Soc}(_RR)$ is essential in $R_R$, then $R$ is a quasi-Frobenius ring and it is also a right $q$-ring.